Adaptive Neuro-Fuzzy Inference System for Drought Forecasting

ORIGINAL PAPER

Adaptive Neuro-Fuzzy Inference System for drought forecasting

Ulker Guner Bacanli Mahmut Firat Fatih Dikbas

Published online: 24 October 2008

Springer-Verlag 2008

Abstract Drought causes huge losses in agriculture and

has many negative influences on natural ecosystems. In this

study, the applicability of Adaptive Neuro-Fuzzy Inference

System (ANFIS) for drought forecasting and quantitative

value of drought indices, the Standardized Precipitation

Index (SPI), is investigated. For this aim, 10 rainfall

gauging stations located in Central Anatolia, Turkey are

selected as study area. Monthly mean rainfall and SPI

values are used for constructing the ANFIS forecasting

models. For all stations, data sets include a total of 516 data

records measured between in 1964 and 2006 years and data

sets are divided into two subsets, training and testing.

Different ANFIS forecasting models for SPI at time scales

112 months were trained and tested. The results of ANFIS

forecasting models and observed values are compared and

performances of models were evaluated. Moreover, the

best fit models have been also trained and tested by Feed

Forward Neural Networks (FFNN). The results demon-

strate that ANFIS can be successfully applied and provide

high accuracy and reliability for drought forecasting.

Keywords Drought forecasting ANFIS Droughtindices Central Anatolia Turkey

1 Introduction

Drought is a threatening global and local problem that has

many damages in various ways. It causes huge losses in

agriculture and has many negative influences on natural

ecosystems. Drought causes degradation of soils and

desertification (Nicholson et al. 1990; Pickup 1998), social

alarm and famine and impoverishment. Studies on climate

change also drew attention to drought in recent years (Byun

and Wilhite 1999) and many studies were made to analyze

the spatial patterns of drought risk in order to assist agri-

cultural or environmental management (Dracup et al.

1980). The development of drought monitoring plans is

a priority in many of these studies (Wilhite 1997; Hayes

et al. 1999) and drought prediction is the subject of some

other studies that investigate the atmospheric causes of

droughts. Drought risk analysis aiming at improving tech-

niques for drought prediction and management are based

on the spatial variation of drought and are mainly focused

on the magnitude, duration, intensity and spatial extent of

droughts. Currently, indirect characteristic features of soil

moisture time series namely drought indices are widely

used. Spatial and temporal extent and severity of drought

can be determined by the help of these indices (Palmer

1995; McKee et al. 1993; Edwards and Mckee 1997; Hayes

1997; Guttmann 1998; Hayes 2000). The Standardized

Precipitation Index (SPI), developed by McKee et al.

(1993), is an effective drought index which has several

advantages over the others. Calculation of the SPI is easier

than the more complex indices such as the Palmer Drought

Severity Index (PDSI; Palmer 1965), because the SPI

requires only precipitation data, whereas the PDSI uses

several parameters. The SPI is comparable in both time and

space and it can be calculated for several time scales

(Srdas and Sen 2003; McKee et al. 1995) and it allows thedetermination of duration, magnitude and intensity of

droughts. The SPI identifies various drought types as

hydrological, agricultural or environmental and it has been

extensively used for drought analysis of many areas of the

world. Several studies focused on the SPIs calculation

U. G. Bacanli M. Firat (&) F. DikbasCivil Engineering Department, Faculty of Engineering,

Pamukkale University, 20017 Denizli, Turkey

e-mail: [email protected]

123

Stoch Environ Res Risk Assess (2009) 23:11431154

DOI 10.1007/s00477-008-0288-5

procedures, which identify the most appropriate frequency

distributions (Guttmann 1998), the effect of time scales on

the parameters (Ntale and Gan 2003), and spatial and

temporal comparability (Keyantash and Dracup 2002).

However, the SPIs spatial stability and coherence in

relation to time scales have not been analysed. Mishra et al.

(2008) investigated the distribution of drought interval

time, mean drought interarrival time, joint probability

density function and transition probabilities of drought

events using the alternative renewable process and run

theory in the Kansabati River basin in India. For this aim,

the Standardized Precipitation Index (SPI) series were

employed and the time interval of SPI was found to have a

significant effect of the probabilistic characteristics of

drought. Mishra and Desai (2005) used the linear stochastic

models known as ARIMA and multiplicative Seasonal

Autoregressive Integrated Moving Average (SARIMA)

models to forecast droughts based on the procedure of

model development. The models were applied to forecast

droughts using standardized precipitation index (SPI) series

in the Kansabati river basin in India. Cancelliere et al.

(2007) proposed two methodologies for the seasonal fore-

casting of SPI, under the hypothesis of uncorrelated and

normally distributed monthly precipitation aggregated at

various time scales. In the first methodology, the auto-

covariance matrix of SPI values was analytically derived,

as a function of the statistics of the underlying monthly

precipitation process. In the second methodology, SPI

forecasts at a generic time horizon M were analytically

determined, in terms of conditional expectation, as a

function of past values of monthly precipitation. The

results showed that the proposed methodologies can be

applied for drought monitoring system. Hughes and

Saunders (2002) used monthly SPIs at time scales of 3, 6,

9, 12, 18, and 24 months for characterizing the drought

climatology of Europe. Bonaccorso et al. (2003) used the

SPI for drought analysis in Italy and Loukas et al. (2004)

applied the SPI for drought forecasting in Greece. Vicente-

Serrano and Lopez-Moreno (2005) analyzed the usefulness

of different SPI time scales to monitor droughts in river

discharges and reservoir storages. The objective was to

determine the most adequate time scales of SPI to monitor

droughts in two basic water usable sources: river dis-

charges and reservoir storages. They found that Time

scales of SPI longer than 12 months do not seem useful to

monitor any drought type in their study areas. Moreira et al.

(2006) analyzed the SPI with the 12-month time scale

through adjusting loglinear models to the probabilities of

transitions between the SPI drought classes.

The new techniques such as artificial neural networks

(ANN), Fuzzy Logic (FL) and ANFIS have been recently

accepted as an efficient alternative tool for modeling

of complex hydrologic systems and widely used for

forecasting. Some specific applications of ANN to

hydrology include modeling rainfall-runoff process (Jeong

and Kim 2005; Kumar et al. 2005; Rajurkar et al. 2004),

hydrologic time series modeling (Jain and Kumar 2007),

sediment concentration estimation (Nagy et al. 2002), esti-

mation of heterogeneous aquifer parameters (Mantoglou

2003), runoff and sediment yield modeling (Agarwal et al.

2006). Morid et al. (2007) examined the utility of ANN

approach for medium and long-term forecasting of both

the likelihood of drought events and their severity. Mishra

and Desai (2006) applied the feed-forward recursive

neural network and ARIMA models for drought fore-

casting using standardized precipitation index (SPI) series

as drought index. The results have demonstrated that

neural network method can be successfully applied for

drought forecasting. Wu et al (2008) applied the neural

network method to establish a risk evaluation model of

heavy snow disaster using back-propagation artificial

neural network (BP-ANN). According to results, BP-ANN

model showed an advantage in heavy snow risk evalua-

tion in Xilingol compared to the conventional method.

Moreover, ASCE Task Committee reports (2000) did a

comprehensive review of the applications of ANN in the

hydrological forecasting context. On the other hand,

several studies have also been carried out using FL in

hydrology and water resources planning (Mahabir et al.

2000; Liong et al. 2000; Nayak et al. 2005; Altunkaynak

et al. 2005). In recent years, Adaptive Neuro-Fuzzy

Inference System (ANFIS), which is integration of ANN

and FL methods, has been used in the modeling of non-

linear engineering and water resources problems (Chang

and Chang 2006; Nayak et al. 2004; Sen and Altunkaynak

2006; Firat 2007; Firat and Gungor 2007, 2008). More-

over, Chou and Chen (2007) have used the neuro fuzzy

computing technique for the development of drought early

warning index. For this aim, an approach has been pro-

posed to develop drought early warning index (DEWI) for

southern Taiwan to detect the drought in advance for

setting up proper plans to mitigate the water shortage

impact.

Drought forecasting plays an important role in the

mitigation of impacts of drought on water resources systems.

Because SPI is one of the most widely used methods

related to drought, accurate and reliable estimation of SPI

is very important. Traditional methods like regression

analysis and autoregressive moving average models

are commonly used in the estimation of hydrological

processes. Moreover FL and ANN methods offer real

advantages over conventional modeling especially when

the underlying physical relationships are not fully under-

stood. FL is employed to describe human thinking and

reasoning in a mathematical framework. The main problem

with FL is that there is no systematic procedure to define

1144 Stoch Environ Res Risk Assess (2009) 23:11431154

123

the MF parameters and to design of fuzzy rules. The con-

struction of the fuzzy rule necessitates the definition of

premises and consequences as fuzzy sets.

In this paper, Adaptive Neuro-Fuzzy Inference System

(ANFIS), which is an integration of ANN and FL methods,

is proposed as an alternative to the traditional methods for

drought forecasting using SPI for multiple time scales. The

main contribution of ANFIS method is that it eliminates the

basic problems in fuzzy modeling (defining the member-

ship function parameters and design of fuzzy ifthen rules)

by using the learning capability of ANN for automatic

fuzzy rule generation and parameter optimization. To

illustrate the applicability of ANFIS method in drought

forecasting, 10 rainfall gauging stations located in Central

Anatolia, Turkey are selected as study area. Monthly mean

precipitation and SPI values are used for constructing the

ANFIS forecasting models. The best fit forecasting model

structure was determined by comparing the forecasted and

observed values.

2 Standard Precipitation Index (SPI)

Standard Precipitation Index calculation is based on long-

term precipitation data. SPI is obtained by dividing the

difference between precipitation and mean to standard

deviation in a specific duration (McKee et al 1993). SPI is a

dimensionless index that takes negative values in drought

periods and positive values in wet periods. The magnitude,

length and duration of drought can be calculated with SPI.

The calculation of SPI is complex because the precipitation

does not fit normal distribution for the periods of

12 months and less and for this reason the precipitation

series are fitted to normal distribution

SPI xi xir

: 1

SPI permits to determine the rarity of a drought or an

anomalously wet event at a particular time scale for any

location that has a precipitation record. A drought event is

considered to occur at a time when the value of SPI is

continuously negative and end when SPI becomes positive

(Mishra et al. 2008). The classes according to the SPI index

are given in the Table 1.

The following steps are applied in the SPI method:

1. Monthly precipitation data sets are organized for a

period of at least 30 years. Different time steps are

determined like 3, 6, 9, 12, 24 or 48 months to monitor

the variations of the indices by considering the

influence of precipitation deficit on various resources.

The time steps may vary according to the condition of

water resources in the area. In the proposed study,

estimation models were constructed with ANFIS

method by using the SPI outputs for 1, 3, 6, 9 and

12 months.

2. Then Gamma distribution is fitted to the data set and

thus the observed precipitation probabilities are

defined. Gamma distribution is the best fitting distri-

bution to the climatologic time series. Gamma

distribution is defined by either the frequency distri-

bution or the probability density function

g x 1baC a x

a1ex=b for x [ 0: 2

a([0) is the shape parameter; b([0) is the scale parameter;x([0) is the precipitation amount, and C (a) is the Gammafunction. In the calculation of a and b, maximumprobability solutions are used. According to this:

a 14A

1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1 4A3

r

!

3

b xa

4

A ln x P

ln x n

5

3. These probability definitions obtained from the present

data may later be used to determine the cumulative

probability of a value observed at any month. In this

situation, the cumulative probability distribution

function is defined as follows:

G x Z

x

0

g x dx 1baC a

Z

x

0

xa1ex=bdx 6

4. Gamma function is undefined for x = 0 and

precipitation distribution can have zero values. When

this is the case, the cumulative probability distribution

is defined as follows:

H x q 1 q G x 7

In the equation above, n is the number of precipitation

observations, q represents the probability for zero value. If

m is used for denoting the zero values in a precipitation

series then the following definition can be made: q = m/n.

Table 1 Classification according to the SPI values

SPI Drought category

2[ Extremely wet1.991.5 Very wet

1.491.0 Moderately wet

0.99(-0.99) Near normal

(1.0)(-1.49) Moderately dry

(1.5)(-1.99) Severely dry

2\ Extremely dry

Stoch Environ Res Risk Assess (2009) 23:11431154 1145

123

5. The cumulative probability value H(x) is converted to

Z variable with a standard normal random value

denoting the SPI value having zero mean value and

variance equal to 1. H(x) is the value of SPI.

Normalization of SPI values enables the consideration

of the variations of precipitation series of that station

by both time and place (McKee et al. 1993; Guttmann

1998).

3 Adaptive Neuro Fuzzy Inference System (ANFIS)

The FL approach proposed by Zadeh (1965) is based on the

linguistic uncertainly expression rather than numerical

uncertainty. FL approach has become popular and has

been successfully used in various engineering problems

(Mahabir et al. 2000; Liong et al. 2000; Nayak et al. 2005;

Sen 2001). Fuzzy inference system (FIS) is a rule based

system consisting of three conceptual components. These

are: (1) a rule-base, containing fuzzy if-then rules, (2) a

data-base, defining the Membership Function (MF) and (3)

an inference system, combining the fuzzy rules and pro-

duces the system results (Firat and Gungor 2007, 2008; Sen

2001). The main problem with fuzzy logic is that there is

no systematic procedure to define the membership function

parameters and to design of fuzzy rules. In recent years,

ANFIS method, which is integration of ANN and FL

methods, has the potential to capture the benefits of both

these methods in a single framework. ANFIS eliminates the

basic problem in fuzzy system design (defining the mem-

bership function parameters and design of fuzzy ifthen

rules) by effectively using the learning capability of ANN

for automatic fuzzy rule generation and parameter opti-

mization. There are two types of FISs, Sugeno-Takagi FIS

and Mamdani FIS, in literature. In this study, Sugeno-

Takagi FIS is used for drought forecasting. The most

important difference between these systems is definition of

the consequent parameter. The consequence parameter in

Sugeno FIS is either a linear equation, called first-order

Sugeno FIS, or constant coefficient, zero-order Sugeno FIS

(Jang et al. 1997). It is assumed that the FIS includes two

inputs, SPI(t - 1) and P(t - 1) and one output, SPI(t). The

membership functions and the structure of are shown in

Fig. 1. For the first-order Sugeno-Takagi FIS, typical two

rules can be expressed as:

Rule 1: IF SPIt 1 is A1 and Pt 1 is B1THEN f1 p1 SPIt 1 q1 Pt 1 r1

Rule 2: IF SPIt 1 is A2 and Pt 1 is B2THEN f2 p2 SPIt 1 q2 Pt 1 r21Input notes (Layer 1) Each node in this layer generates

membership grades of the crisp inputs and each nodes

output O1i is calculated by:

O1i lAi SPIt 1 for i 1; 2;O1i lBi2Pt 1 for i 3; 4

8

where, SPI(t - 2) is the SPI value at time (t - 2) to the

node i, P(t - 1) is the actual precipitation at time (t - 1),

A1 B1 B1Membership Degree

B2

P (t-1) Rule Base and Inference System

Membership Degree

A2

SPI (t-2)

1111 )1(*)1(* rtPqtSPIpf ++=

2222 )1(*)1(* rtPqtSPIpf ++=

P (t-1)

SPI (t-2)

)1(),1((1 tPtSPIw

N

N

)1(),1((2 tPtSPIw

11 fw

22 fw

Layer 1 Layer 2 Layer 3 Layer 4 Layer 5

21

11 ww

ww +=

21

22 ww

ww

+=

1A

2A

1B

2B

)1(),1((1 tPtSPIf

)1(),1((1 tPtSPIf

)1(),1((2 tPtSPIf

Fig. 1 The scheme of AdaptiveNeuro-Fuzzy Inference System


123

SPI(t) is the SPI value at time (t) to the node i, Ai and Bi are

the linguistic labels, pi, qi and ri are the consequence

parameters, lAi and lBi are the MFs for Ai and Bi linguisticlabels, respectively and in this study, the Gauss MF is used,

as

O1i lAix eSPIt1c2

2r2 : 9

Rule nodes (Layer 2) The outputs of this layer, called

firing strengths O2i , are the products of the corresponding

degrees obtained from layer 1, named as w as follows:

O2i wi lAi SPIt 1lBi Pt 1; i 1; 2 10

Average nodes (Layer 3) Main target is to compute the

ratio of firing strength of each ith rule to the sum firing

strength of all rules. The firing strength in this layer is

normalized as

O3i wi wiP

i wii 1; 2 11

Consequent nodes (Layer 4) The contribution of ith rule

towards the total output or the model output and/or the

function defined is calculated by Eq. (12)

O4i wifi wipi SPIt 1 qiPt 1 ri i 1; 212

Output nodes (Layer 5) This layer is called as the output

nodes in which the single node computes the overall output

by summing all incoming signals

Q5i f SPIt 1;Pt 1 X

i

wi fi wif1 wif2

P

i wifiP

i wi14

where wi is the ith node output from the previous layer as

demonstrated in the third layer. ANFIS applies the hybrid-

learning algorithm, which consists of the combination of

the gradient descent and the least-squares methods to

determine the input and output model parameters. The task

of the learning algorithm is to tune all the antecedent and

consequence parameters to make the ANFIS response

match the training data. The gradient descent method is

used to assign the nonlinear antecedent parameters and the

least-squares method is employed to identify the linear

consequent parameters. All these parameters are updated

using this hybrid learning algorithm until acceptable error

is reached. The details and mathematical background of

these algorithms can be found in Jang et al. (1997) and in

Nayak et al. (2004).

4 Study area and data

The temperature difference between summer and winter is

high, the precipitation generally occurs in spring and winter

and dry periods dominate summers. This climate is experi-

enced in Central, East, Southeast Anatolia and Trakya

region. Climate of Central Anatolia has the following

properties: The weather in the summer is a little hot and

winters are cold. The severity of cold weather increases

towards the east parts of Central Anatolia. Natural flora

consists of steppes in the lower regions and dry forests in the

higher regions because of summer droughts. Mean temper-

ature of January, the coldest month, is 0.7C and it is 22C inJuly, the hottest month. Annual mean temperature is 10.8C.Mean annual precipitation is 413.8 mm and most of the

precipitation occurs in winter and spring seasons. The per-

cent of summer rains among the annual total is 14.7%. The

annual mean proportional moisture in the region is 63.7%.

Observed monthly rainfall data records from ten meteoro-

logical stations (Aksaray, Ankara, Cankr, Eskisehir,Karaman, Kayseri, Konya, Krsehir, Nevsehir and Yozgat)located in Central Anatolia, Turkey, have been selected for

this study. The length of available records at these stations is

between 1964 and 2006. The SPI for this study have been

calculated on the basis of these rainfall data.

5 Drought forecasting by ANFIS

5.1 Input variables

Different time steps like 3, 6, 12, 24 and 48 months are

determined as (1) for monitoring the variations in the

indexes by considering the effect of precipitation lack on

different water resources. In this study, the values of SPI

and precipitation in the previous months are used for

generating a drought estimation model with ANFIS

f x; y w1f1 w2f2w1 w2

w1 SPIt 1;Pt 1f1SPIt 1;Pt 1 w2SPIt 1;Pt 1f2SPIt 1;Pt 1w1SPIt 1;Pt 1 w2SPIt 1;Pt 1

13


123

method. For this, the SPI outputs for 1, 3, 6, 9 and

12 months were considered. In the construction of esti-

mation models, again, different models were generated for

each of the SPI output for 1, 3, 6, 9 and 12 months. The

data sets for all stations were divided into two subsets,

training and testing data set. The training data set includes

data records measured between 1964 and 1986 years. In

order to get more reliable evaluation and comparison,

models are tested by evaluating a data set which was not

used during the training process. Testing data set consists

of data records observed between 1987 and 2006 years.

The statistical parameters, minimum value, maximum

value, mean, standard deviation, variance, skewness coef-

ficient and Kurtosis for training and testing data sets are

calculated and given in Tables 2 and 3 to see a comparison

of the training and testing data sets.

5.2 Model structures

One of the most important steps in developing a satisfac-

tory forecasting model is the selection of the input

variables. Because, these variables determine the structure

of forecasting model and affect the weighted coefficient

and the results of the model. Here, different estimation

models were constructed for each phase. The models for 1,

3, 6, 9 and 12 months were named as SPI-1, SPI-3, SPI-6,

SPI-9 and SPI-12, respectively. Here, SPI-1, SPI-3 and

SPI-6 were considered as the index for short term or sea-

sonal variation, SPI-9 for short term drought and SPI-12

was considered as the drought index for long term. 20

models with different input numbers and structures were

constructed for each phase by using these variables. In this

study, forecasting models based on various combinations

of antecedent values of actual precipitations and SPI values

were constructed (Table 4). In each model every input

variable must be clustered into several class values in layer

1 to build up fuzzy rules. And each fuzzy rule would be

constructed through several parameters of membership

function in layer 2. As the number of parameters increases

with the fuzzy rule increment, the model structure becomes

more complicated. In this study, the subtractive fuzzy

clustering function was used to establish the fuzzy rule

based on the relationship between the inputoutput vari-

ables. In order to determine the nonlinear input and linear

output parameters, the hybrid algorithm was used. The

Table 2 The statistical parameters for training data sets (19641986)

Min. Max. Mean SD Variance Skewness Kurtosis

Aksaray 0.0 110.1 28.96 23.17 536.98 0.775 0.142

Ankara 0.0 121.5 34.86 26.08 680.54 0.799 -0.005

Cankr 0.0 137.7 34.62 26.64 709.96 1.012 0.856

Eskisehir 0.0 128.2 34.07 25.92 671.92 0.954 0.949

Karaman 0.0 144.1 29.15 26.88 722.91 1.182 1.469

Kayseri 0.0 133.2 30.50 23.70 562.01 0.927 0.869

Konya 0.0 112.2 28.74 23.47 551.11 0.876 0.433

Krsehir 0.0 145.8 31.84 26.49 702.12 0.820 0.430

Nevsehir 0.0 116.7 34.29 26.56 705.89 0.668 -0.107

Yozgat 0.0 192.3 48.79 38.68 1496.22 0.876 0.475

Table 3 The statistical parameters for testing data sets (19872006)

Min. Max. Mean SD Variance Skewness Kurtosis

Aksaray 0.0 101.3 28.47 23.21 539.14 0.813 0.100

Ankara 0.0 122.4 32.33 25.92 672.28 0.953 0.630

Cankr 0.0 149.8 32.76 27.46 754.28 1.381 2.175

Eskisehir 0.0 129.7 29.16 22.36 500.28 1.329 2.823

Karaman 0.0 121.8 26.27 22.60 510.87 0.956 0.802

Kayseri 0.0 164.7 33.62 27.56 760.07 1.078 1.663

Konya 0.0 124.0 24.94 23.87 570.18 1.540 2.909

Krsehir 0.0 121.0 31.39 25.42 646.24 0.916 0.764

Nevsehir 0.0 148.8 33.91 28.14 792.11 1.053 1.318

Yozgat 0.0 172.0 50.01 37.61 1414.92 0.819 0.245

Table 4 The structures of forecasting models

Model Input structure Output

M1 SPI(t - 1) SPI(t)

M2 SPI(t - 1), SPI(t - 2) SPI(t)

M3 SPI(t - 1), SPI(t - 2), SPI(t - 3) SPI(t)

M4 SPI(t - 1), SPI(t - 2), SPI(t - 3), SPI(t - 4) SPI(t)

M5 SPI(t - 1), SPI(t - 2), SPI(t - 3), SPI(t - 4),SPI(t - 5)

SPI(t)

M6 SPI(t - 1), SPI(t - 2), SPI(t - 3), SPI(t - 4),SPI(t - 5), SPI(t - 6)

SPI(t)

M7 R(t - 1) SPI(t)

M8 R(t - 1), R(t - 2) SPI(t)

M9 R(t - 1), R(t - 2), R(t - 3) SPI(t)

M10 R(t - 1), R(t - 2), R(t - 3), R(t - 4) SPI(t)

M11 R(t - 1), R(t - 2), R(t - 3), R(t - 4), R(t - 5) SPI(t)

M12 R(t - 1), R(t - 2), R(t - 3), R(t - 4), R(t - 5),R(t - 6)

SPI(t)

M13 SPI(t - 1) R(t - 1) SPI(t)

M14 SPI(t - 1), SPI(t - 2) R(t - 1) SPI(t)

M15 SPI(t - 1), SPI(t - 2) R(t - 1), R(t - 2) SPI(t)

M16 SPI(t - 1), SPI(t - 2), SPI(t - 3) R(t - 1) SPI(t)

M17 SPI(t - 1), SPI(t - 2), SPI(t - 3) R(t - 1),R(t - 2)

SPI(t)

M18 SPI(t - 1), SPI(t - 2), SPI(t - 3), SPI(t - 4)R(t - 1)

SPI(t)

M19 SPI(t - 1), SPI(t - 2), SPI(t - 3), SPI(t - 4)R(t - 1), R(t - 2)

SPI(t)

M20 SPI(t - 1), SPI(t - 2), SPI(t - 3), SPI(t - 4),SPI(t - 5) R(t - 1)

SPI(t)


123

learning procedure and the construction of the rules were

provided by this algorithm. The performance of ANFIS

models for training and testing data sets were evaluated

according to statistical criteria such as, Correlation Coef-

ficient (CORR), Efficiency (E), and Root Mean Square

Error (RMSE). The CORR is a commonly used statistic

and provides information on the strength of linear rela-

tionship between the observed and the computed values.

The E is one of the widely employed statistics to evaluate

model performance. The values of CORR and E close to

1.0 indicate good model performance. The RMSE statistic

indicates a models ability to predict a value away from the

mean.

As it is impossible to show the model results for each

phase having 20 models because of space restrictions, only

the results for SPI-6 at Ankara station (Ankara is the capital

of Turkey and it is one of the cities where water shortage

and drought is severely experienced) are presented. The

testing performances of ANFIS models for SPI-6 are given

in Fig. 2.

When the results of the ANFIS models are compared, it

is seen that the performances of models composed of

precipitation values belonging to the previous time step are

lower than the performances of the other models. The

results of models in which SPI is used, show that the

performances of the models at all stations are close to

each other and that the model defined as M5 has a better

performance than the others. A general decrease in per-

formance was observed in all models when the values at

(t - 6) time step were used. When the results of the models

consisting of only the precipitation values are evaluated, it

is seen that M11 is the model with the best performance for

all stations. It was also observed that the model (M7)

composed of precipitation values at the (t - 1) time step

has the lowest performance. The figure shows that the

model (M12) generated by using the values at (t - 6) time

step generally have lower performances. On the other side,

the investigation of the results given in the graphs for SPI-6

show that the models generated with the previous values of

SPI and precipitation data have a better performance. By

using the precipitation and SPI variables together for all

stations, an improvement has been achieved in the model

performances. According to the criteria, the model defined

as M20 ANFIS for Aksaray, Ankara, Karaman, Kayseri,

Krsehir, Konya and Yozgat stations, had the best resultsover the other models. On the other hand, while the best

results are obtained from the M5 model for Eskisehir and

Cankr stations, it was determined that M14 ANFIS modelhad the best performance for Nevsehir station. As a result,

the performances of the best fit ANFIS models for SPI-6 at

all stations (after the analysis of all stations, only perfor-

mances of the models giving the most suitable results are

presented.) are shown in Table 5.

It can be stated that the model performances of ANFIS

models for all stations are at an acceptable level for SPI-6.

Figure 3 shows the performances of ANFIS models at

Ankara station for the data from 1 month to 12 months

(SPI-1 to SPI-12). In this figure, the variations of CORR, E

and RMSE criteria for SPI-1 to SPI-12 at Ankara station

during the testing period are demonstrated.

The results of other stations that are not presented here

due to space restrictions indicate that the ANFIS models

for SPI-12 have shown the best performance at all sta-

tions. It is seen that the performances of ANFIS model for

SPI-12 at Ankara station is better than those of other

models. The values of CORR and E of ANFIS models for

SPI-1 are lower than those of other models. The reason

for the ANFIS models developed by using SPI outputs of

12 months to show a better performance is that the SPI

Ankara Station

0.21 2 3 4 5 6 7 8 9 10 11 12 13 14 15 1716 18 19 20 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 1716 18 19 20

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Model

CO

RR

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

E

CORRE

Ankara Station

0.5

0.6

0.7

0.8

0.9

1.0

1.1

Model

RM

SE

RMSE

Fig. 2 Comparison ofperformances of ANFIS Models

for SPI-6 at Ankara station

Table 5 The performances of the best fit models for SPI-6 at allstations

Station Testing set Training set

CORR E RMSE CORR E RMSE

Aksaray (M20) 0.837 0.686 0.628 0.810 0.656 0.514

Ankara (M20) 0.824 0.685 0.549 0.876 0.754 0.526

Cankr (M5) 0.773 0.599 0.644 0.870 0.767 0.508

Eskisehir(M5) 0.825 0.710 0.547 0.879 0.781 0.471

Karaman (M20) 0.826 0.714 0.578 0.860 0.741 0.521

Kayseri (M20) 0.846 0.712 0.601 0.855 0.731 0.462

Krsehir (M20) 0.804 0.642 0.615 0.828 0.694 0.541

Konya (M20) 0.815 0.68 0.603 0.872 0.761 0.443

Nevsehir (M14) 0.841 0.701 0.608 0.810 0.710 0.470

Yozgat (M20) 0.818 0.667 0.578 0.821 0.704 0.549


123

values calculated for a long term include dry and wet

periods for longer duration. Short term periods like 1 or

3 months may include a wet or a dry period for a short

time. For example, in 3 months period, drought occurs

more frequently and for a shorter time and when the

period increases the duration of drought increases but its

frequency decreases. This means that for shorter periods

the SPI values may contain 1 month dry and 1 month wet

period and this causes instability. Passages between

positive and negative values occur more frequently and

this also results with instability. For this reason, the

ANFIS estimation models constructed with the SPI values

calculated for shorter periods, cannot catch dry and wet

periods and give unsuccessful results. Besides, the SPI

outputs for 12 months have a more stable run. Thus, the

ANFIS models developed by using SPI outputs for

12 months can catch dry and wet periods and give better

results. Figure 4 shows the results of ANFIS models for

Ankara station from SPI-1 to SPI-12.

In order to evaluate the results of ANFIS models, the

best fit models for Ankara station (SPI-1 to SPI-12) have

also been tested by Feed Forward Neural Networks

(FFNN) and Multiple Linear Regression (MLR). The

FFNN models have been trained and tested using the same

data sets. The error back propagation algorithm and tangent

activation function is used for training/testing of the FFNN

models. The number of hidden layers and the hidden

neurons in this layer, the learning rate, the coefficient of

momentum and epochs were selected by trial and error

method during the training. The results of FFNN and MLR

models for SPI-12 at Ankara station are shown in Table 6

and Fig. 5.

Comparing performances of ANFIS models for Ankara

station, it is seen that the performance of the ANFIS model

for SPI-12 are better than other ANFIS models for SPI-1 to

SPI-9. As a result, it is said that ANFIS can be successfully

applied and provide high accuracy and reliability for

drought forecasting. On the other hand, comparing the

results of ANFIS and FFNN forecasting models for Ankara

station (SPI-1 to SPI-12), it can be seen that the RMSE

values of the ANFIS models are lower than that of FFNN

model. In addition, the values of E and CORR of the

ANFIS model are also higher than those of FFNN models.

The results suggest that the ANFIS method is superior to

the FFNN method in the forecasting of drought. It may be

noted that a trial and error procedure has to be performed

for FFNN models to develop the best network structure

while such a procedure is not required in developing an

ANFIS model. Figure and table indicate that the best result

was obtained from the models developed for SPI-12 as

in the ANFIS method. Comparing the performances of

ANFIS and MLR models, it can be seen that the values of

E and CORR of the ANFIS model are also higher than

those of MLR models. The NRMSE values of ANFIS

model are also lower than those of MLR models. The

results suggest that the ANFIS method is also superior to

the MLR method in the drought forecasting. The results

show that ANFIS method can be successfully applied to

establish accurate and reliable drought forecasting models.

6 Conclusions

SPI is one of the most widely used methods related to

drought and SPI should be estimated accurately and reli-

ably. Traditional methods like regression analysis and

autoregressive moving average models are commonly used

in the estimation of hydrological processes.

In this paper, Adaptive Neuro-Fuzzy Inference System

(ANFIS) was proposed as an alternative drought forecast-

ing tool to the traditional methods. The main contribution

of ANFIS method is that it eliminates the basic problems

in fuzzy modeling (defining the membership function

parameters and design of fuzzy ifthen rules) by using

the learning capability of ANN for automatic fuzzy rule

generation and parameter optimization.

To illustrate the applicability of ANFIS method in

drought forecasting, 10 rainfall gauging stations located in

Central Anatolia, Turkey were selected as study area.

Different ANFIS forecasting models for SPI-1, SPI-3,

SP-6, SPI-9 and SPI-12 were trained and tested. When the

results of the ANFIS models are compared, it is seen that

only the performances of models composed of precipita-

tion values belonging to the previous time step are lower

Ankara Station

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1 3 6 9 12

Month

CO

RR

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

E

CORRE

Ankara Station

0.2

0.4

0.6

0.8

1

1.2

1 3 6 9 12

Month

RM

SE

RMSE

Fig. 3 The performances ofANFIS models for SPI-1, SPI-3,

SPI-6, SPI-9 and SPI-12 at

Ankara station


123

Ankara (SPI-1)

-6.0

-4.0

-2.0

0.0

2.0

4.0

1 14 27 40 53 66 79 92 105 118 131 144 157 170 183 196 209 222 235

MonthSP

I

ForecastedObserved

-4

-2

0

2

4

-4 -2 0 2 4

Observed

For

ecas

ted

Ankara (SPI-3)

-4.0

-2.0

0.0

2.0

4.0

1 14 27 40 53 66 79 92 105 118 131 144 157 170 183 196 209 222 235

Month

SPI

ForecastedObserved

-4

-2

0

2

4

-4 -2 0 2 4

Observed

For

ecas

ted

Ankara (SPI-6)

-4.0

-2.0

0.0

2.0

4.0

1 14 27 40 53 66 79 92 105 118 131 144 157 170 183 196 209 222 235

Month

SPI

ForecastedObserved

-4

-2

0

2

4

-4 -2 0 2 4

Observed

For

ecas

ted

Ankara (SPI-9)

-4.0

-2.0

0.0

2.0

4.0

1 14 27 40 53 66 79 92 105 118 131 144 157 170 183 196 209 222 235

Month

SPI

ForecastedObserved

-4

-2

0

2

4

-4 -2 0 2 4

Observed

For

ecas

ted

Ankara (SPI-12)

-4.0

-2.0

0.0

2.0

4.0

1 14 27 40 53 66 79 92 105 118 131 144 157 170 183 196 209 222 235

Month

SPI

ForecastedObserved

-4

-2

0

2

4

-4 -2 0 2 4

Observed

For

ecas

ted

Fig. 4 The results of ANFISmodels for for Ankara station

(SPI-1 to SPI-12)


123

than the performances of the other models. The results of

models in which only the SPI is used, show that the

model named M5 has a better performance than the other

models. It was also observed that when the SPI value at

(t - 6) time step is used, there is a decrease in perfor-

mance for all stations generally. On the other hand, when

the results of models containing only the precipitation

values were investigated, it was found that M11 has

shown the best performance for all stations. The model

defined as M7 composed of the precipitation value at time

step (t - 1) had the lowest performance. The results

indicate that the models (M12) generated by using the

precipitation values at time step (t - 6) generally have a

lower performance. By using the precipitation and SPI

variables together for all stations, an improvement was

achieved in the model performances. According to the

Table 6 Comparison of performances of ANFIS, FFNN and MLR models for Ankara station

Station Testing set Training set

CORR E RMSE CORR E RMSE

M20 ANFIS (for SPI-1) 0.392 0.371 1.016 0.573 0.502 0.968

M20 ANFIS (for SPI-3) 0.490 0.422 0.707 0.584 0.569 0.654

M20 ANFIS (for SPI-6) 0.824 0.685 0.549 0.876 0.754 0.526

M20 ANFIS (for SPI-9) 0.851 0.733 0.507 0.920 0.847 0.401

M20 ANFIS (for SPI-12) 0.893 0.808 0.425 0.930 0.865 0.375

M20 FFNN (for SPI-1) 0.314 0.298 1.254 0.487 0.451 1.026

M20 FFNN (for SPI-3) 0.417 0.402 0.916 0.561 0.524 0.845

M20 FFNN (for SPI-6) 0.752 0.625 0.652 0.833 0.694 0.575

M20 FFNN (for SPI-9) 0.813 0.674 0.577 0.858 0.738 0.525

M20 FFNN (for SPI-12) 0.851 0.722 0.512 0.887 0.794 0.453

M20 MLR (for SPI-1) 0.306 0.257 1.291 0.380 0.302 1.354

M20 MLR (for SPI-3) 0.411 0.398 1.096 0.576 0.528 0.885

M20 MLR (for SPI-6) 0.719 0.584 0.669 0.829 0.687 0.582

M20 MLR (for SPI-9) 0.804 0.600 0.621 0.894 0.799 0.462

M20 MLR (for SPI-12) 0.811 0.593 0.619 0.904 0.822 0.435

Ankara FFNN Model (SPI-12)

-4.0

-2.0

0.0

2.0

4.0

1 14 27 40 53 66 79 92 105 118 131 144 157 170 183 196 209 222 235

Month

SPI

ForecastedObserved

-4

-2

0

2

4

-4 -2 0 2 4

Observed

For

ecas

ted

Ankara MLR Model (SPI-12)

-4.0

-2.0

0.0

2.0

4.0

1 14 27 40 53 66 79 92 105 118 131 144 157 170 183 196 209 222 235

Month

SPI

ForecastedObserved

-4

-2

0

2

4

-4 -2 0 2 4

Observed

For

ecas

ted

Fig. 5 The results of FFNN andMLR models for SPI-12 at

Ankara station


123

criteria given in the figures, the model defined as M20

ANFIS model, which consists of the combination of the

antecedent values of the rainfall and SPI variables, for

Aksaray, Ankara, Karaman, Kayseri, Krsehir, Konya andYozgat stations, had the best results over the other

models. Moreover, while the best results are obtained

from the M5 ANFIS model, which includes the anteced-

ent values of SPI variable, for Eskisehir and Cankrstations, it was determined, that M14 ANFIS model had

the best performance for Nevsehir station. Comparing the

performances of ANFIS models for SPI-1, SPI-3, SP-6,

SPI-9 and SPI-12 at 10 stations during the testing period,

it was seen that the performances of the models for SPI-

12 at all stations are better than those of other models.

The reason for the ANFIS models to show a better per-

formance is that the SPI values calculated for long

periods contain longer periods of dry and wet periods.

This means that for shorter periods the SPI values may

contain 1-month dry and 1-month wet period and this

causes instability. Passages between positive and negative

values occur more frequently and this also results with

instability. For this reason, the ANFIS estimation models

constructed with the SPI values calculated for shorter

periods, cannot catch dry and wet periods and give

unsuccessful results. Besides, the SPI outputs for

12 months have a more stable run. Thus, the ANFIS

models developed by using SPI outputs for 12 months can

catch dry and wet periods and give better results. In order

to evaluate the results of ANFIS models, the best fit

models for Ankara station (SPI-1, SPI-3, SP-6, SPI-9 and

SPI-12) have also been trained and tested by FFNN

method. The FFNN models have been trained and tested

using the same data sets. Comparing the results of ANFIS

and FFNN forecasting models for Ankara station, it can

be seen that the RMSE values of the ANFIS models were

lower than that of FFNN model. In addition, the values of

E and CORR of the ANFIS model were also higher than

those of FFNN models. To get more reliable evaluation of

performance of ANFIS model, the best fit models for

Ankara station were compared to MLR model. It can be

seen that the NRMSE value of ANFIS models were lower

than those of MLR models. The values of E and CORR

of ANFIS models were also higher than those of MLR

models.

The results suggest that the ANFIS method is superior to

the FFNN and MLR methods in the forecasting of drought.

Moreover, the result showed that ANFIS method can be

successfully applied to establish accurate and reliable

drought forecasting models.

Acknowledgments The authors are grateful for editors and anon-ymous reviewers for their helpful and constructive comments on an

earlier draft of this paper.

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http://dx.doi.org/10.1007/s00477-007-0194-2http://dx.doi.org/10.1007/s00477-007-0181-7

Adaptive Neuro-Fuzzy Inference System for drought forecastingAbstractIntroductionStandard Precipitation Index (SPI)Adaptive Neuro Fuzzy Inference System (ANFIS)Study area and dataDrought forecasting by ANFISInput variablesModel structures

ConclusionsAcknowledgmentsReferences

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